Eddy Interaction Model for Turbulent Suspension in Reynolds-Averaged

Advances in Water Resources 111 (2018) 435–451

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Advances in Water Resources

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Eddy interaction model for turbulent suspension in Reynolds-averaged T Euler–Lagrange simulations of steady sheet ﬂow ⁎ Zhen Cheng ,1,a, Julien Chauchatb, Tian-Jian Hsua, Joseph Calantonic a Civil and Environmental Engineering, University of Delaware, Newark, DE 19716, USA b Laboratory of Geophysical and Industrial Flows (LEGI), BP 53, Grenoble Cedex 9 38041, France c Marine Geosciences Division, U.S. Naval Research Laboratory, Stennis Space Center, MS 39529, USA

ARTICLE INFO ABSTRACT

Keywords: A Reynolds-averaged Euler–Lagrange sediment transport model (CFDEM-EIM) was developed for steady sheet Euler–Lagrange model flow, where the inter-granular interactions were resolved and the flow turbulence was modeled with a low Eddy interaction model Reynolds number corrected kω− turbulence closure modified for two-phase flows. To model the effect of Turbulent suspension turbulence on the sediment suspension, the interaction between the turbulent eddies and particles was simulated Steady sheet flow with an eddy interaction model (EIM). The EIM was first calibrated with measurements from dilute suspension Rouse profile experiments. We demonstrated that the eddy-interaction model was able to reproduce the well-known Rouse Sediment transport rate profile for suspended sediment concentration. The model results were found to be sensitive to the choice of the

coefficient, C0, associated with the turbulence-sediment interaction time. A value C0 = 3 was suggested to match the measured concentration in the dilute suspension. The calibrated CFDEM-EIM was used to model a steady sheet flow experiment of lightweight coarse particles and yielded reasonable agreements with measured velocity, concentration and turbulence kinetic energy profiles. Further numerical experiments for sheet flow sug-

gested that when C0 was decreased to C0 < 3, the simulation under-predicted the amount of suspended sediment in the dilute region and the Schmidt number is over-predicted (Sc > 1.0). Additional simulations for a range of Shields parameters between 0.3 and 1.2 conﬁrmed that CFDEM-EIM was capable of predicting sediment transport rates similar to empirical formulations. Based on the analysis of sediment transport rate and transport layer thickness, the EIM and the resulting suspended load were shown to be important when the fall parameter is less than 1.25.

1. Introduction The conventional single-phase-based sediment transport models assume the dynamics of transport can be subjectively separated into bedload Studying sediment transport in rivers and coastal regions is critical and suspended load (e.g., van Rijn, 1984a; 1984b). While the suspended to understand the fluvial geomorphology, loss of wetland, and beach load are directly resolved, the bedload are parameterized by empirical erosion. For example, significant engineering efforts were devoted to formulations. Several laboratory measurements of sheet flow with the control the river discharge and sediment budget to reduce the loss of full profile of sediment transport flux and net transport rate indicated Louisiana wetland (Allison et al., 2012; Mossa, 1996). In the Indian that the split of bedload and suspended load may be too simple because River inlet, significant erosion of the north beach is mitigated through sediment entrainment/deposition is a continuous and highly dynamic proper beach nourishment that interacts with littoral drift process near the mobile bed (e.g., O’Donoghue and Wright, 2004; Revil- (Keshtpoor et al., 2013). The characteristics of sediment transport vary Baudard et al., 2015). In sheet flow, the two prevailing mechanisms significantly with sediment properties and flow conditions, and it is driving the sediment transport are inter-granular interactions and tur- widely believed that sheet flow plays a dominant role in nearshore bulent suspension (Jenkins and Hanes, 1998; Revil-Baudard et al., beach erosion and riverine sediment delivery, especially during storm 2015). In order to model the full profile of sediment transport, both and flood conditions, respectively. mechanisms must be taken into account. In the past decade, many Sheet flow is an intense sediment transport mode, in which a thick Eulerian two-phase flow models have been developed for sheet flow layer of concentrated sediment is mobilized above the quasi-static bed. transport in steady (Jenkins and Hanes, 1998; Longo, 2005; Revil-

⁎ Corresponding author. E-mail address: [email protected] (Z. Cheng). 1 Now at Applied Ocean Physics & Engineering, Woods Hole Oceanographic Institution, Woods Hole, MA 02543, USA. https://doi.org/10.1016/j.advwatres.2017.11.019 Received 15 April 2017; Received in revised form 31 October 2017; Accepted 19 November 2017 Available online 21 November 2017 0309-1708/ © 2017 Elsevier Ltd. All rights reserved. Z. Cheng et al. Advances in Water Resources 111 (2018) 435–451

Baudard and Chauchat, 2013) and oscillatory flows (Dong and Zhang, a Lagrangian approach. 2002; Hsu et al., 2004), Amoudry et al.,(Chen et al., 2011; Cheng et al., In the stochastic Lagrange model for particle dispersion, the tur- 2017a; Liu and Sato, 2006). By solving the mass and momentum bulent agitation to the sediment particles are considered either through equations of fluid phase and sediment phase with appropriate closures a random-walk model (RWM) or an eddy interaction model (EIM). In for interphase momentum transfer, turbulence, and intergranular the RWM framework, the strength of particle velocity fluctuations are stresses, these models are able to resolve the entire profiles of sediment typically assumed to be similar to the fluid turbulence, and a series of transport without the assumption of bedload and suspended load. random velocity fluctuations are directly added to the particle velo- In the continuum description of the sediment phase, the assumption cities. While the Lagrange model with RWM is successful in studying of uniform particle properties and spherical particle shapes are usually the particle dispersion in mixing layer (Coimbra et al., 1998) and dilute adopted. To better capture the polydisperse nature of sediment trans- suspension (Shi and Yu, 2015), the assumption of estimating the par- port and irregular particle shapes, the Lagrangian approach for the ticle velocity fluctuations based on the fluid turbulence is crucial, and particle phase, namely the Discrete Element Method (DEM, Cundall and many researchers found that the correlation between the particle and Strack, 1979; Maurin et al., 2015; Sun and Xiao, 2016a) is superior to fluid fluctuations are highly dependent on the particle Stokes number, the Eulerian approach because individual particle properties may be St= tpl/ t (Balachandar and Eaton, 2010), where tp is the particle re- uniquely specified (Calantoni et al., 2004; Fukuoka et al., 2014; Harada sponse time, and tl is the characteristic time scale of energetic eddies. and Gotoh, 2008). One of the main challenges in modeling sheet flow For the particles with very small inertia (St ≪ 1), they can closely follow arise from the various length scales involved in inter-granular interac- the eddy motion. However, if St ≫ 1, the particle trajectory is hardly tions and sediment-turbulence interactions. To resolve the flow turbu- affected by the fluid eddy motion. Due to the particle inertia effect, it lence and turbulence-sediment interactions in sheet flow, the compu- was found that the fluid turbulent intensity needs to be enhanced for tational domain needs to be sufficiently large to resolve the largest medium to coarse particles (Shi and Yu, 2015). This problem can be eddies, while the grid resolution should be small enough to resolve the largely remedied by the EIM (Matida et al., 2004), where the fluid energy containing turbulent eddies. This constrain becomes even more velocity fluctuations associated with the fluid turbulence are added challenging in the Euler–Lagrange modeling framework. Large domains through the particle-sediment interaction force, i.e., the drag force. This require both a large number of grid points to resolve a sufficient approach incorporates the particle inertia effect naturally and it is ap- amount of turbulence energy cascade (i.e., large-eddy simulation) and a plicable for a wide range of sediment properties. Graham (1996) found large number of particles in a given simulation (e.g., Finn et al., 2016). that the dispersion of inertial particles may be correctly represented It is well-established that in sheet flow, the transport layer thickness with a suitable choice of maximum interaction time and length scales scales with the grain size and the Shields parameter (Wilson, 1987), with the eddies. This model was later improved by using a randomly suggesting that a common sheet flow layer thickness must be about sampled eddy interaction time, in which more realistic turbulent scales several tens of grain diameters. To simulate the largest eddies and their become possible, and the enhanced dispersion of high-inertia particles subsequent cascade, the domain lengths in the two horizontal directions are captured. In the previous studies of particle dispersion (e.g., Shi and must be proportional to the boundary layer thickness, which is usually Yu, 2015), the turbulent intensity is either prescribed from the em- about several tens of centimeters. For a bed layer thickness of 50 grains pirical formula, or modeled using clear fluid turbulence closure without with a typical grain diameter of 0.2 mm, sheet flow simulations may considering turbulence modulation by the presence of particles. In sheet require at least several tens of millions of particles. Therefore, to effi- flow, it is well-known that the sediment-turbulence interaction is im- ciently model sediment transport for many scenarios in sheet flow, a portant in attenuating the flow turbulence, thus the presence of sedi- turbulence-averaged approach for the carrier phase may be necessary. ment can dissipate/enhance flow turbulence through drag/density In a turbulence-averaged formulation, turbulent eddies are not resolved stratification. and their effects on the averaged flow field are often parameterized via In this paper, we present an application of the eddy interaction eddy viscosity. In this case, the domain lengths in the two horizontal model (EIM) in a Reynolds-averaged Euler–Lagrange formulation to directions are solely determined by the largest length scale of inter- study sheet flow. The eddy interaction model is implemented into an granular interaction, which is usually captured within 50 grain dia- open source coupled CFD-DEM scheme called CFDEM (Goniva et al., meters (Maurin et al., 2015). Consequently, the number of particles 2012), and the new solver is called CFDEM-EIM. The fluid phase is needed for each sheet flow simulation is limited to no more than a few modeled in a similar way as the Eulerian two-phase flow model Sed- hundred thousand. FOAM (Cheng et al., 2017a), and the particles are modeled with the With a goal to develop a robust open-source coupled Computational discrete particle model, LIGGGHTS (Kloss et al., 2012). The paper is Fluid Dynamics-Discrete Element Method (CFD-DEM) for sheet flow organized in the following manner. The model formulation is described applications, we adopt a turbulence-averaged approach in this study. in Section 2. The model calibration with dilute suspension experiments Existing Reynolds-averaged CFD-DEM models have the capability to is presented in Section 3.1, followed by model validation of steady sheet model bedload transport (Durán et al., 2012; Maurin et al., 2015) and flow (Section 3.2) using a comprehensive dataset (Revil-Baudard et al., sheet flow for coarse sand (Drake and Calantoni, 2001), where the 2015; 2016). Section 4 discusses the model sensitivity of the resulting inter-granular interactions are dominant, and the turbulent suspension sediment diffusivity and Schmidt number to model coefficients in the is of minor importance. The previous studies made significant pro- eddy interaction scheme, and effects of the EIM on the modeled sedi- gresses in understanding the sediment dynamics due to intergranular ment transport rate and transport layer thickness are also evaluated. collisions and interactions with the mean flow, and the key character- Finally, a practical regime for the EIM to be important is proposed istics such as sediment transport rate and transport layer thickness close based on the fall parameter. Concluding remarks are given in Section 5. to the empirical formulations were obtained. In more energetic sheet flows with medium to fine sand particles, the role of turbulence-induced suspension can become important, where substantial sediment 2. Model formulations suspension occurs above the bedload layer (Bagnold, 1966; Sumer et al., 1996). In such condition, a more complete closure models for 2.1. Discrete particle model turbulent suspension and turbulence modulation by particles are needed. The natural way of describing the diffusion and dispersion of In the framework of the discrete element method (Cundall and dispersed particles is to sample the turbulent velocity statistics along Strack, 1979), the position of each particle is tracked by integrating the their trajectories in a stochastic manner (Taylor, 1922), and this idea particle equation of motion, lays the foundation of modeling the turbulent motions of particles with

436 Z. Cheng et al. Advances in Water Resources 111 (2018) 435–451

dxpi, into the Reynolds-averaged component ufi, and the turbulent fluctu- = vi, ′ dt (1) ating component u f, i. In CFDEM-EIM, the Reynolds-averaged velocities are provided by the carrier fluid model. While the turbulent fluctuating where x is the position of particle i and v is the translational velocity. p, i i component is modeled with an additional eddy-interaction closure (see The governing equation for the translational motion of particle i with Section 2.4). To generalize the drag coefficient for both spherical and radius ri and mass mi may be written as, non-spherical particles, the drag coefficient CD is given by (Haider and Nc Levenspiel, 1989), dvi mi =+fffgpf, i ∑ ().nij,, ++ tijm i dt j=1 (2) ⎡ 24 B C ⎤ CfϕD = () (1++ARe ·p ) , ⎢ Re 1/+ DRe⎥ The forces acting on the ith particle include the particle-fluid interac- ⎣ p p ⎦ (7) tion force, f , the gravitational force, m g, and the normal, f , and pf, i i n, ij where Reϕpf=−(1 )uv,i −i dνi / fis the particle Reynolds number, νf is tangential, ft, ij, contact forces where Nc is the number of particles in the fluid kinematic viscosity, and di is the diameter of the spherical contact with the particle i. The rotational motion of particle i with particle or an equivalent sphere that has the same volume as the non- moment of inertia Ii may be written as, spherical particle. The four parameters A, B, C, and D are proposed to η Nc be functions of particle sphericity, , with dΩi Ii =+∑ (),MMtij,, rij dt Aηη=−+exp(2.3288 6.4581 2.44862 ), (8) j=1 (3) Bη=+0.0964 0.5565 , where Ωi is the angular velocity of particle i. The torque acting on (9) particle i from particle j consists of two components. Closures are used Cηηη=−+−exp(4.905 13.8944 18.422223 10.2599 ), (10) for Mt, ij, which is generated by the tangential force, and Mr, ij, which is commonly known as the rolling friction torque (Luding, 2008). Dηηη=+−+exp(1.4681 12.2584 20.732223 15.8855 ). To model grain contact forces, we adopt the soft-sphere approach (11) (Cundall and Strack, 1979) based on Hertz–Mindlin theory. Hertz For spherical particles, η = 1, and for nonspherical particles, η <1. theory is implemented in the normal direction, and the improved In the drag coefficient (Eq. (7)), a correction for particle concentration, Mindlin no-slip model is implemented in the shear direction f (),ϕ is introduced to take into account the hindered settling effect (Mindlin, 1949). In the soft-sphere model (e.g., Renzo and Maio, 2005), (Felice, 1994), particles are allowed to overlap slightly, and the contact between two 2−n particles may be described as a nonlinear spring-dashpot, where the f ()ϕϕ=− (1 ) . (12) δ normal contact force, fn, ij, is determined by the overlap, ij, and relative where, the empirical exponent, n, is related to the particle Reynolds velocity between colliding particles, Vr, ij, while the tangental force, ft, number, ij, is calculated in a similar way and includes the tangental contact 2 history. In addition, if the tangential force exceeds the Coulomb fric- (1.5− logRep ) n =−3.7 0.65 exp⎡− 10 ⎤. ⎢ ⎥ tional limit, the particles begin to slide, and the tangential force is set to ⎣ 2 ⎦ (13) μ ffi ftij,,= μc fnij, where c is the Coulomb friction coe cient. In the present study, we only consider the torque induced by particle-particle/par- The local sediment concentration, ϕ , is calculated by averaging the ticle-wall contact, and the influence of fluid-induced torque is ignored. sediment instantaneous sediment concentration within one CFD time step, In general, the particle-fluid interaction force, fpf, is the sum of all fl fl N types of particle- uid interaction forces on individual particles by uid, 1 s including the so-called drag force, fd, the pressure gradient force, fp, ϕ = ∑ ϕj, Ns buoyancy force if assuming locally hydrostatic flow, virtual mass force, j=1 (14) ff fvm, Basset force, fB and lift forces such as the Sa man force, fSaff, and where N is number of DEM time steps within one CFD time step (see fl s the Magnus force, fMag. We assume that the uid and particles share the more details in Section 2.5), the divided volume fraction model fi fl pressure eld, thus the uid pressure gradient force is also included in (Goniva et al., 2012) is used for the instantaneous sediment con- fl the uid-particle interactions (Maxey and Riley, 1983; Zhou et al., centration at each DEM time step, where the particle volumes are di- 2010). In CFDEM-EIM, only the two dominant forces, namely the drag vided into 29 parts using the satellite points, and the volumes are dis- fl force and pressure gradient force, are retained. Here the total uid- tributed into the touched fluid grid cells. The model works well when particle interaction force acting on particle i may be written as, particle size is similar to the mesh size. fpf,,, i=+ff d i p i. (4) 2.2. Fluid phase governing equations The pressure gradient force acting on particle i is calculated as, fl fpi,,=−∇()·f xi ipV i, (5) In contrast of the particle phase, the uid phase is solved in an Eulerian framework and the coupled Euler–Lagrange system follows the fl ∇ where fx, i is the external body force driving the steady ow. ip is the so-called model “A” (e.g., Zhou et al., 2010). By further carrying out fl interpolated uid pressure gradient at particle i, and Vi is the volume of Reynolds-averaging, the fluid momentum equation may be written as, particle i. The drag force acting on particle i is expressed as, ∂−ρϕf (1 )uf 1 +∇·[ρϕf (1 − )uuff ] = (1 − ϕ )f x − (1 − ϕp ) ∇ fdi,,,,=−−CA D siuvuv f i i() f i i , ∂t 2 (6) +∇·(1),τgFf +ρϕf − + d (15) where uf ,i is the instantaneous fluid velocity interpolated at particle i, and As, i is the projected area of the ith spherical particle (or equivalent where the overbar ‘’ denotes the ensemble-averaged fields, ρf is the fluid spherical particle for non-spherical particles). According to the Rey- density. The first term on the right-hand-side (R.H.S.) is the external nolds decomposition, the instantaneous fluid velocity is decomposed body force that drives the steady flow. The second term on R.H.S. is the

437 Z. Cheng et al. Advances in Water Resources 111 (2018) 435–451

pressure gradient force. τf is the total fluid stress tensor, which includes Table 1 List of coefficients in the LRN kω− equations for two-phase flows. the viscous stress (τν) and the Reynolds stress (τft). The last term on the fl R.H.S. is the sum of the drag force from the particles within the uid α σ σ 0 α0* Rek Reω Reβ Cμ k ω Cs C1ω C2ω C3ω C4ω grid volume (Vcell), which must satisfy the Newton’s 3rd law,

N N 1/9 0.024 6 2.95 8 0.09 2.0 2.0 1.0 0.52 0.072 0.14 0 1 s cell Ffd =− ∑∑ di, . NV scellj==11i (16) The sediment concentration (ϕ ) calculated directly by grid aver- where the operation ‘:’ denotes the scalar product of two tensors. C* is aging in the DEM is usually not smooth, and averaging errors may μ model coefficients with low Reynolds number corrections based on the depend on the averaging length scale (Simeonov et al., 2015). To ensure original coefficient Cμ (see Table 1), numerical stability, a diffusion model is often used to obtain a sufficiently smoothed concentration profile, 4 4/15+ (Retβ / Re ) CC* = , μ μ 4 ∂ϕ 1(/)+ Retβ Re (22) =∇·(Dϕt ∇ ). ∂t (17) where the model constant Re = 8 is a critical Reynolds number. The diffusion constant, D , is calculated as, DLdt= 2/,where L is a β t t d d Except for the last two terms on the R.H.S. of Eq. (21), the rest of the length scale, and dt is the fluid phase time step (i.e., CFD time step). The terms in the present k equation are essentially the same as those in the choice of length scale, Ld= is found to be stable and necessary when f d clear fluid TKE equation. The last term in Eq. (21) represents the the fluid grid length is similar to or smaller than the particle diameter buoyancy term. For typical sediment concentration with an upward (Capecelatro and Desjardins, 2013; Pirker et al., 2011). Note that this decaying profile, this term represents the well-known sediment-induced smoothed concentration field is only used in the fluid governing density stratification that can attenuate the fluid turbulence. The fourth equations and turbulence closures. The model results (mainly in term on the R.H.S. represents attenuation of TKE due to drag with β Sections 3 and 4) of the sediment concentration, sediment velocity and calculated as, transport rate are directly obtained from the DEM part (i.e., not smoothed by the diffusion model). To ensure a stable calculation of 3 ρCf D Ur conservation of mass, a mixture continuity equation for the in- β = , 4 d (23) compressible fluid-particle system can be derived and is solved (Cheng et al., 2017a), where CD is calculated by Eq. (7) with particle Reynolds number,

∇−·[(1ϕϕ )uufs + ] = 0. (18) Reϕdνpf=−(1 )Ur / , in which |Ur| is the magnitude of relative velocity seen by the fluid. Here, to better estimate Ur in dilute condition, where instantaneous sediment concentration fluctuation is significant, a 2.3. Fluid turbulence modeling temporal average of the relative velocity is carried out,

As brieﬂy described in Eq. (15), the total ﬂuid stress tensor consists 1 t Ur = ()uufs− dt, ∫t of the viscous stress (τν) and the Reynolds stress (τft): tt− 0 0 (24)

⎡ T 2 2 ⎤ where t is the starting time of the time average, and t is the current run τττfνft=+=ρϕννf (1 − ) (fftf+ )⎛∇+∇− u ufff Iu ∇· ⎞ − k I, 0 ⎣ ⎝ 3 ⎠ 3 ⎦ time of the simulation. For a steady sheet flow application, this time (19) average procedure is representative of the ensemble-averaged relative fl in which, the Reynolds stress in the Reynolds-averaged Eulerian fluid velocity between uid and sediment phases. Throughout the simula- model may be written as, tions in this study, the quasi-steady state is usually reached within 5 s of numerical simulations, thus we choose t0 = 5 s. To quantify the effect of ⎡ T 2 2 ⎤ fluid-particle turbulence modulation, the parameter λ is introduced by τuuIuIft =−ρϕνf (1 ) ft⎛∇+∇− f fff ∇· ⎞ − k, ⎣ ⎝ 3 ⎠ 3 ⎦ (20) following (Cheng et al., 2017a), where I is an identity tensor, ∇T is the transpose of gradient tensor, ν is ft λ = e−CSts· , (25) the turbulent eddy viscosity, and kf is the fluid turbulent kinetic energy (TKE). The eddy viscosity and TKE are modeled with a low Reynolds where Cs is an empirical coefficient. St= tpl/ t , is the particle Stokes number version k − ω turbulence model (LRN k − ω closure number, i.e., the ratio of the particle response time (tp = ρ /β) to the fi fl s (Wilcox, 1992)) modi ed for two-phase ows. characteristic time scale of energetic eddies. In the literatures of Rey- nolds-averaged turbulence closures, the general expression for the eddy fl 2.3.1. Low Reynolds number corrected k − ω closure for two-phase ow life time can be written as, tltμf= CCω/( ), and the value of the coeffi- In LRN k − ω closure, the low Reynolds number correction was cient Ct ranges from 0.135 to 0.41 (Milojeviè, 1990), which is highly introduced based on the local Reynolds number, Rekνωtfff= /( ). With dependent on the flow conditions. From the preliminary numerical this correction, the LRN k − ω closure is capable of capturing transi- experiment, we found the eddy life time is vital for the turbulence- tional turbulent flow in the near-bed region. To take into account of the sediment interaction, thus we chose the coefficient Ct = 1/6 by fol- ff fl – sediment e ect on the ow turbulence, the sediment turbulence in- lowing (Cheng et al., 2017a), and the model coefficients associated with fl teraction terms were added to both the transport equations for the uid the eddy life time are left as model calibration. For example, the TKE (k ) and specific turbulent dissipation frequency (ω ), similar to the f f coefficient Cs in Eqn. (25) is calibrated using the sheet flow experi- approach suggested by Amoudry (2014) and Chauchat et al. (2017), mental dataset (see Section 3.2) to match the flow hydrodynamics, and k ν it was chosen to be Cs = 1. ∂ f τft ⎡⎛ ft ⎞ ⎤ +∇=∇+∇uff·:·k uff⎜⎟ν + ∇kCkωf − * ff The balance equation for ω follows the original work of ∂t ρ ⎢ σ ⎥ μ f f ⎣⎝ k ⎠ ⎦ Wilcox (1992). However, for turbulence-particle sinteractions, similar ω 2(1βλϕk− ) f 1 νft ⎛ ρ ⎞ damping terms as in the kf equation are included. The f equation is − − s − 1·g ∇ϕ , ρϕ(1− ) (1− ϕ ) σ ⎜ρ ⎟ written as, f c ⎝ f ⎠ (21)

438 Z. Cheng et al. Advances in Water Resources 111 (2018) 435–451

Table 2 List of numerical simulations of dilute sand suspension in steady channel ﬂows.

3 3 Cases d (mm) ρs(kg/m ) ws(cm/s) u*(cm/s) Φ ×10 N

Kiger and Pan (2002) 0.195 2605 2.4 2.85 0.23 476 NS1 in Muste et al. (2005) 0.23 2650 2.4 4.2 0.46 803

∂ω ω τ ν where κ = 0.41 is the von Karman constant, and the bottom frictional f f ft ⎡⎛ ft ⎞ ⎤ 2 +∇=uff· ωC* :· ∇+∇uff⎜⎟ν + ∇ωCωfωf− 2 ∂t 1ω kρ ⎢ σ ⎥ velocity is calculated as u =+∂∂()/ννuzat the wall boundary. It f f ⎣⎝ ω ⎠ ⎦ * fftf was found that this formulation of bottom boundary condition for 2(1βλϕω− ) f ωf 1 νft ⎛ ρs ⎞ − C34ω − C ω ⎜ − 1·⎟g ∇ϕ smooth wall is robust for low to high Reynolds number turbulent ρϕ(1− ) kϕf (1− ) σc ρ f ⎝ f ⎠ boundary layer flows. ωbed + Γ(ϕ ), On the other hand, the bed is covered with a thick layer of sediment Trelax particles in sheet flow condition, and the particles imposes a rough wall (26) boundary to the flow above the bed. However, the location of the bed in where the fourth and fifth terms take into account of the sediment effect sediment transport is difficult to determine as a priori due to possible on the fluid turbulence through drag and buoyancy, respectively. The erosion processes. To avoid this complexity, the last term on the R.H.S. ffi coe cients C1*ω is also modulated using the local turbulent Reynolds of Eq. (26) is proposed to impose a desired value of specific turbulence number as, dissipation rate, ωbed, in the sediment bed, and Γ(ϕ ) is a step-like function of sediment concentration, 1 αReRe0 + tω/ CC1*ω = 1ω , α* 1/+ Retω Re (27) tanh[500(ϕϕ−+ )] 1 Γ(ϕ ) = b , 2 (32) where α* is a damping function based on Ret,

where ϕb should be specified as the sediment concentration in the bed, αReRe0* + tk/ α* = . so that the ωf value is only imposed inside the sediment bed. In this 1/+ Retk Re (28) study, we choose ϕb = 0.55. An intrinsic relaxation timescale is used for ffi where α0* and Rek are model coe cient for the low Reynolds number Trelax, which sums the proper timescale on the R.H.S. of the ωf equation, corrections. The model constant C1ω, C2ω, σk, σω,Rek,Reω and α0 are similar to the closure coefficients suggested by Guizien et al. (2003) 1 2(1βλϕ− ) 11νft ⎛ ρs ⎞ =+Cω23ωf C ω + C 4ω − 1·g ∇ϕ . ffi T ρϕ(1− ) kϕ(1− ) σ ⎜ρ ⎟ (see Table 1). The coe cient of the buoyancy term, C4ω = 0 is chosen relax f f c ⎝ f ⎠ fi fl for stable strati cation applicable for steady sheet ow (Rodi, 1987). (33) Through a series of sensitivity test, we found that the modeled flow It shall be noted that the relaxation time scale proposed here is velocities are also sensitive to the coefficient C3ω, and the optimum positive in sheet flow applications. For specific energy dissipation fre- value of C ω is 0.14, which is close to the value 0.2 suggested by 3 ω fi Amoudry (2014). A full list of the coefficients associated with the low quency bed inside the bed, the rough wall value can be speci ed as Reynolds number k − ω model used in this study is presented in (Wilcox, 1988), Table 1. u 2 * Finally, the turbulent eddy viscosity ν is calculated by the fluid ωbed= S r , ft νf (34) turbulence kinetic energy kf (TKE) and specific turbulence dissipation fi rate ωf, where u* is the bottom frictional velocity at the bed interface speci ed based on the flow forcing to drive the steady channel flow and Sr is a kf ναft = * . parameter depending on the bed roughness, ωf (29) 2 ⎧⎛ 200 ⎞ + It shall be noted that the LRN k − ω can be reduced to the original ⎜⎟,5k < ⎪ k + s k − ω model (Wilcox, 1993) in the fully turbulent region when the local ⎪⎝ s ⎠ Sr = , Reynolds number is sufficiently high compared with the critical Rey- ⎨ 2 K 200 K + ⎪ r ⎡⎛ ⎞ r ⎤ (5−ks ) + nolds numbers. + ⎜⎟− ek,5≥ ⎪ kk+++⎢ k ⎥ s ⎩ ss⎣⎝ ⎠ s ⎦ (35) 2.3.2. Smooth and rough wall functions + where kkuνs = s / f is the normalized wall roughness in wall units, and The wall functions for a smooth bed and rough bed are both relevant * ks is the Nikuradse’s equivalent sand roughness, which is related with to the present study. For clear fluid or dilute suspension, such as the the sand grain size, ks = 2.5d. The original coefficient Kr is 100 as experiment of Muste et al. (2005) to be discussed in Section 3.1,a suggested by Wilcox (1988), however, Fuhrman et al. (2010) proposed smooth wall is exposed and the ωf value in the viscous sublayer scales that this coefficient needs to be reduced to Kr = 80 to match the law of with 1/z2, where z is the distance to the bottom wall boundary. As a wall. Therefore, Kr = 80 is used throughout this paper. result, ωf goes to infinity at the wall boundary. In the numerical im- fi ω fi plementation, a nite value of f is imposed to the rst grid above the 2.4. Eddy interaction model solid smooth wall, and the following bottom boundary condition is fi speci ed (Bredberg et al., 2000; Menter and Esch, 2001), The drag force (Eq. (6)) in the particle momentum equation depends 22 on the instantaneous fluid velocity. However, only the Reynolds-aver- ωwall =+ωωvis log , (30) aged fluid velocity (ufi, ) is solved and hence an additional closure fl fl fl ′ with the ωwall value specified as a blend function of the values in the model for the uid velocity uctuation in turbulent ow (u f, i) is re- viscous sublayer (ωvis) and logarithmic layer (ωlog ), quired. Appropriate consideration of particle dispersion by turbulent eddies provides a key suspension mechanism in sediment transport (i.e., 6νf u ωvis ==, ωlog * , turbulent suspension). Following Graham and James (1996), particle 0.075z 2 Cκz o μo (31) dispersion by turbulence can be modeled with a stochastic Eddy

439 Z. Cheng et al. Advances in Water Resources 111 (2018) 435–451

Interaction Model (EIM), and a series of random Lagrangian velocities the Poisson ratio υ as 2(1GυE+= ) . fl t can be used to represent the uid turbulent motions, i.e. u′fi, = Uσ i 1, (3) dts is required to be smaller than the Hertzian contact time in order t t σ vVσ′fi, = i 2, and w′fi, = Wσ i 3, where 1, 2, 3 are Gaussian random num- to capture the contact process. The Hertzian contact time is the bers with a zero mean value and a standard deviation of unity. In this duration of a pair of particles in contact, which can be estimated as, −1 −1 study, the velocity fluctuations are calculated using the fluid turbulent 221/5 11 11 TmrEVcr= 2.87( * / ** n) , where r* =+ , m*= + , t t t ()rrij ( mmij) kinetic energy, UVWi ==i i =2/3 kfi, , where kf, i is interpolated − 2 1 − ν2 1 turbulence kinetic energy at the mass center of particle i. It is possible to 1 − νi j and E* = ⎜⎟⎛ + ⎞ . For a contact between a sphere particle i model the anisotropic velocity fluctuations in three directions, how- ⎝ Ei Ej ⎠ ever, to be consistent with the two-equation turbulence-averaged with wall j, the same relationship applies to E*, whereas rr* = i and models (LRN k − ω closure), the turbulent fluctuations are assumed to mm* = i. be isotropic. fl t t The dt is constant throughout the simulation once appropriately In the eddy interaction model, the velocity uctuations (i.e., Ui , Vi , s t Wi ) are updated every step with the particle position. However, the chosen to satisfy the above criteria, and the particle velocities are up- random numbers σ1, 2, 3 remained unchanged until the eddy interaction dated every dts, where the forces acting on each particles are solved time tI is exceeded, which is determined either when a particle has according to Eq. (4). In the calculation of drag forces, the eddy inter- completely crossed a turbulent eddy or remains in an eddy but exceeds action model is implemented to model the turbulence-induced sediment the eddy life time. The mean life time of the turbulent eddy can be suspensions, where a fluctuating component of velocities are in- −1 estimated as TCωli,,= (6 μ f i ) in the LRN k − ω model. However, the troduced to the drag forces through a stochastic procedure, which is instantaneous turbulent eddy life time is of random-like nature (Kallio outlined as follows: and Reeks, 1989; Mehrotra et al., 1998) and it is estimated as, (a) Initially at t = 0, the time marker tmark, i, and eddy interaction time tei,0=−CξTln(1 − ) li ,, (36) tI, i are set to zero for each particle. σ fl fl where ξ is the random number ranging from 0 to 1. As discussed in (b) Random numbers 1, 2, 3 are generated and the uid velocity uc- Section 2.3, due to the uncertainties in the parameterization of the eddy tuation u′fi, , v′fi, , w′fi, are updated. The drag forces are then calculated life time, the coefficient C0 is introduced as a constant for model cali- using Eq. (6). bration (see Section 3.1). As a result, the turbulent eddy length le can be (c) The following two scenarios are considered: estimated as ltkei,,= ei2/3 f , i . With the estimation of the turbulent eddy (i) If (tt−≥mark,, i) t I i, the particle enters a new turbulent eddy, and σ length le, the eddy crossing time for a particle can be computed as then new Gaussian random number 1, 2, 3 are generated, and (Gosman and Loannides, 1983), fluid fluctuations are updated with the new values of σ1, 2, 3. Both tmark, i and tI, i are updated to the current values. ⎛ lei, ⎞ tci,,=−t piln⎜⎟ 1 − , (ii) Else if (tt−convection terms (including the k − ω equations) are dis- between the fluid phase and sediment phase utilizes the open source cretized using a total variation diminishing (TVD) scheme based on a code CFDEM (Goniva et al., 2012), which couples the Finite-volume Sweby limiter (Sweby, 1984). The second-order central scheme is used CFD toolbox OpenFOAM (Weller, 2002) with the DEM solver for the diffusion terms. For the temporal integration, a first-order im- LIGGGHTS (Kloss et al., 2012). At the beginning of the simulation, the plicit Euler scheme is used. The PISO (Pressure Implicit Splitting Op- particle positions and velocities are initialized in the DEM module, and eration) algorithm is used for the velocity-pressure decoupling, so that the fluid velocity and turbulence quantities are initialized in the CFD the continuity equation (18) is satisfied. More details on the numerical module. The loop of the CFD-DEM coupling begins with the update of solution procedures for the fluid solver can be found in Rusche (2002). particle positions and velocities for Ns DEM time steps within one fluid time step (dt), in which the time step dts in the DEM module is related to the fluid time step by dtdtNs = / s. In the contact force model, the energy 3. Model results stored in the collision increases rapidly with the overlapping length of fi particles, thus the time step dts should be sufficiently small to avoid the Through preliminary numerical experiments, we con rmed that the unphysical energy generation due to particle contacts. In this study, the modeled sediment concentration profile is sensitive to the prediction of fl ffi following three criteria are used to determine dts: uid TKE and the coe cient C0 in estimating the turbulent eddy life time in the eddy interaction model (see Eq. (36)). This is somewhat

(1) The overlap length δn is smaller than 0.5% of particle diameter d, expected as the turbulent intensity and the eddy interaction time are ff i.e., dts < 0.005d/Vrn, where Vrn is normal component of the relative the main factors di erentiating the present stochastic procedure for velocity to the contact face between two contacting particles. modeling turbulent diffusion from incoherent random motions. There- (2) To capture the energy transmission in the solid particles, the time fore, we first validated the turbulence closure with direct numerical fl fl step dts is chosen to be small enough compared with the Rayleigh simulation (DNS) of clear uid turbulent channel ow. After estab- −1 lishing the accuracy of the turbulence closure for clear fluid, the coef- timescale Tr, where TπrρGr =+s/ (0.163 ν 0.8766) and G is the shear modulus. G is further related to the Young’s modulus E and ficient C0 in the eddy interaction model is calibrated with the dilute suspension experiment of Kiger and Pan (2002), where the velocity,

440 Z. Cheng et al. Advances in Water Resources 111 (2018) 435–451 sediment concentration and Reynolds shear stress profiles are measured simulation of Moser et al. (1999). Meanwhile, it is evident that the fl overall shear stress follows a linear profile τu=−2 (1 zh / ) in the for sand in a steady channel ow over a smooth bed (starved bed). The tot * calibrated model is then applied to predict the suspended sand con- range of z/h > 0.1 (dashed curve in Fig. 1b). The modeled TKE is also centration and turbulence of another similar dilute suspension experi- remarkably close to the DNS data. It is evident that the LRN k − ω ment reported by Muste et al. (2005). Because the sediment con- model is able to resolve the peak of turbulent kinetic energy near the centration is very dilute ( < 1%) and there is negligible deposit on the bottom wall (around zh= 0.02 ), even though the peak value from the LRN k − ω closure (4.4u 2) is slightly smaller than the DNS data (4.75u 2). bed, these datasets allow us to solely calibrate the C0 value in the eddy * * interaction model without complication from intergranular interac- Kiger and Pan (2002) later conducted a sediment-laden turbulent tions. After the calibration, the model is applied to the steady sheet flow flow experiment at a similar Reynolds number as Moser et al. (1999). experimental configuration of Revil-Baudard et al. (2015). A sensitivity The data of Kiger and Pan (2002) can be further used to calibrate the C0 ffi analysis of the model results to the C0 value is investigated in detail to coe cient in the EIM. In the experiment, the half channel height is fl illustrate the effects on the turbulent suspension in steady sheet flow. h = 0.02 m, which is the same as the clear uid simulation at Reτ = 570, The capability of the present CFDEM-EIM is further demonstrated by and hence we kept the same domain setup and boundary conditions as comparing predictions of sediment transport rate and transport layer the clear fluid 1DV simulation. In the DEM implementation, the parti- thickness with classical empirical formula. cles are tracked in a meshless 3D domain (domain size is the same as in the CFD). The lateral boundaries in DEM are periodic, while the wall boundary was used for both the top and bottom boundaries to conserve 3.1. Model calibration of dilute suspension in steady channel flow the number of particles in the domain. The sediments are spherical particles with a density of ρ = 2605 kg/m3, and the grain diameter is Firstly, the LRN k − ω turbulence closure is validated against the s d = 0.195 mm. The particle settling velocity is about 0.024 m/s, which DNS dataset of Moser et al. (1999) for a clear fluid steady wall-bounded corresponds to a shape factor η = 1 in the drag model (see channel flow at a Reynolds number of Reuhν==/ 570 (Moser et al., τf* Eqs. (8)–(11)). The domain averaged sediment volumetric concentra- 1999), where h is the channel half-width. We carried out a 1DV nu- tion in the experiment is Φ2.310=×−4. To match the domain-aver- merical simulation at the same Reynolds number with a vertical domain aged sediment concentration in Kiger and Pan (2002), a total of height h = 0.02 m. A shear-free symmetric boundary condition is used N = 476 particles are simulated in the DEM. at the top boundary, while the bottom boundary condition is a no-slip To calibrate the C0 value in the EIM, we carried out four simulations wall. The standard smooth wall functions for k and ω are used at the with different C0 values, C0 = 1, 2, 3, 4. The resulting profiles of the bottom wall boundary (see Eq. 31). In both x and y directions, periodic streamwise velocity, TKE and sediment concentration are compared boundaries are used and only one grid cell is used in these two direc- with the measured data of Kiger and Pan (2002) and clear fluid DNS tions with a grid size (domain size) of LL==0.02 m. The vertical xy data of Moser et al. (1999) in Fig. 2. Due to the dilute sediment con- domain is discretized into 168 grid cells with a uniform grid size centration in the domain, the numerical results of the streamwise ve- Δz = 0.122 mm. The flow is driven by a mean pressure gradient of locity profile are not very sensitive to the C0 values, so only the velocity fx = 43.5 Pa/m, so that the bottom frictional velocity is u = 0.0285 m/ * profiles corresponding to C0 = 3 is shown. We notice that the measured s. The distance of the first grid center to the bottom boundary patch velocity profile differs slightly from the DNS data, possibly due to the corresponds to a wall unit Δ+ ==0.5uνΔ/ 1.76. Therefore, the first z * zf effect of the presence of sediment in the water column and/or mea- cell center is within the viscous sublayer. surement uncertainties. In addition, the averaged particle velocity The comparisons of the mean Reynolds shear stress, velocity and profile (not shown) is very close to the fluid velocity. The measured TKE profiles between the LRN k − ω model and DNS data are shown in data of the turbulent intensity is only available for the streamwise (u′ ) Fig. 1. Very good agreements on all three profiles are obtained, espe- rms and vertical (w′ ) velocity fluctuations. In order to compare the tur- cially the velocity profile and Reynolds shear stress. The agreement in rms bulence kinetic energy of numerical results and measured data, the the Reynolds shear stress profile confirms that the flow has reached a spanwise velocity fluctuation is reconstructed following the quasi-steady state and the flow condition is similar to the DNS

Fig. 1. The comparison of non-dimensiona- fi lized (a) streamwise velocity pro le (uf /u*), (b) Reynolds shear stress profile (− uw′′/ u2), and (c) TKE profile (k /u 2) ff * f * between LRN kω− closure (solid curves) and DNS data (symbols) of Moser et al. (1999). In panel (b), the dashed curve denotes a linear fit of the total shear stress, τu=−2 (1 zh / ). tot *

441 Z. Cheng et al. Advances in Water Resources 111 (2018) 435–451

Fig. 2. The comparison of (a) fluid velocity profile, (b) nondimensional fluid turbulent kinetic energy (k , normalized by u 2) and (c) f * normalized concentration profile between model results (solid curves) and measured (or DNS) data (symbols). In all panels, the triangle symbols are measured data from Kiger and Pan (2002), and the DNS data of Moser et al. (1999) is shown as circle symbols in panel (a) and (b). In panel (c), sediment concentrations are plotted in semi- logarithmic scale. The solid curves corresponds to C0 = 1 (magenta), C0 = 2 (blue), C0 = 3 (red) and C0 = 4 (black). The dashed curve is the fitted Rouse profile with Ro = 1.44. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

relationship suggested by Jha and Bombardelli (2009), which correspond to a shape factor of η = 0.644 (see Eqs. (8)–(11)). A vuurms′ =−0.3 0.6 rms′ . Thus the turbulent kinetic energy in the experi- similar numerical setup as the simulation of Kiger and Pan (2002) is * 22 2 ment can be estimated by kuf = ()/rms′ + v rms′ + w rms′ 2. The model results used, except that the domain height is h = 0.021 m to match the ex- also predict slightly smaller turbulence kinetic energy compared with perimental condition. The streamwise and spanwise domain lengths are clear fluid counterpart, but the reduction is very small due to dilute specified to be LLxy==100 d. In the vertical direction, uniform grid sediment concentration. Overall, the velocity and turbulence kinetic sizes are used with Nz = 210 grids to resolve the entire flow depth, and energy profiles are in good agreement with the measured data. the first grid center above the bottom wall corresponds to a wall unit fi ff + The sediment concentration pro les corresponding to di erent C0 Δz = 1.05. A total number of particles used in the DEM is N = 803, values are presented in Fig. 2c (solid curves) and they can be compared which matches the domain averaged concentration Φ4.610=×−4. with measured data (symbols in Fig. 2c). It is evident that the sus- The model results of velocity profile, concentration profile and TKE pended sediment concentration is strongly affected by the coefficient (kf) profile with C0 = 3 are compared with the measured data in Fig. 3. fl fi C0. In general, more significant sediment suspension is obtained with a The resulting uid velocity pro le (Fig. 3a) matches the measured data larger C0 value. Clearly, a C0 value of 1 under-predicts the suspended reasonably well, except that the velocity magnitude is slightly over- sediment concentration, and almost all the sediment particles accu- predicted in the range of 0.1 < z/h < 0.5. The normalized sediment mulate near the bottom (z/h < 0.15, see magenta curve in Fig. 2c). concentration (normalized by the mean concentration ϕr at the re-

When the C0 value is increased to C0 = 2, considerably more sediments ference location zhr/0.= 1) shows that the suspended sediment con- are suspended, however, the resulting sediment concentration remains centration profile is similar to the measured data as well as the Rouse fi to be lower than the measured data. The optimum C0 value is found to pro le with a Rouse number Ro = 0.86 (dashed curve in Fig. 3b). In be C0 = 3, and the resulting sediment concentration profile is in good Fig. 3c, the numerical result of TKE is compared with the measured agreement with the measured data. Finally, using C0 = 4 clearly over- data. The measured data of the turbulent intensity is reconstructed in predicts sediment concentration. It is well-known that the sediment the same way as the measurement of Kiger and Pan (2002). Overall, the concentration profile in a steady turbulent channel flow follows the magnitude of the turbulent kinetic energy is smaller than the measured Rouse profile (Vanoni, 2006), data by no more than 30%. However, the vertical profile shape is re- produced well by the model. −Ro ϕ ⎛ hz− zr ⎞ In summary, the LRN k − ω model is validated using a clear fluid = ⎜⎟, ϕr ⎝ z hz− r ⎠ (38) DNS dataset of Moser et al. (1999) and the eddy interaction model is calibrated by using the measurements from Kiger and Pan (2002) and where RowScκu= s /( ) is the Rouse number, in which the Schmidt * Muste et al. (2005). It is found that the optimum C0 value that matches number Sc is the ratio of turbulent eddy viscosity over the sediment the measured concentration profiles for both experiments is C0 = 3.0, diffusivity. z is the reference location above the bed, and ϕ is the r r while C0 < 3.0 underestimated the suspended sediment concentration. concentration at the reference location. We choose the reference loca- Therefore, this calibrated C0 value is applied to the sheet flow appli- tion to be zhr = 0.1 , corresponding to the lowest elevation that the cations in the following subsection. Reynolds shear stress follows a linear profile. The dashed curve in Fig. 2c shows the fitted Rouse profile to the measured data with the Rouse number Ro = 1.44. It is evident that both the measured data and 3.2. Steady sheet flow the numerical result with C0 = 3 match the Rouse profile very well. The calibrated C0 is further applied to another similar dilute sus- In this section, we further apply CFDEM-EIM to model steady sheet pension experiment reported by Muste et al. (2005, see Table 2). The flow, where both bedload (inter-granular interaction dominant) and flow is driven by a prescribed pressure gradient force in order to match suspended load (turbulent suspension dominant) are important. The fl the bottom friction velocity of u* = 0.042 m/s in a ow depth of laboratory experiments reported by Revil-Baudard et al. (2015); 2016), 3 fl fi “ ” h = 0.021 m. The sand density is ρs = 2650 kg/m and the grain dia- which include a steady ow over a rough xed bed ( FB ) and a steady meter is d = 0.23 mm. The measured settling velocity is about 2.4 cm/s, sheet flow (mobile bed, “MB”) were used for model validation. The flow

442 Z. Cheng et al. Advances in Water Resources 111 (2018) 435–451

Fig. 3. The comparison of (a) fluid velocity profile, (b) normalized concentration profile, and (c) fluid turbulent kinetic energy (k , normalized by u 2) between model re- f * sults and measured data of case NS1 in Muste et al. (2005). In all panels, the symbols represent the measure data in Muste et al. (2005), and the solid curves are model results. In panel (b), sediment concentrations are plotted in semi-log scale. The dashed curve is the Rouse profile with Ro = 0.86.

Table 3 retained in the DEM contact model. In the DEM model, the Young’s Flow condition and sediment properties in the fixed bed (“FB”) and mobile bed (“MB”) modulus of particles is specified as E =×5106 Pa, the restitution sheet flow experiment of Revil-Baudard et al. (2015,2016). ffi ffi coe cient is e = 0.5, the Coulomb friction coe cient is μc = 0.5 and the ρ 3 ν 2 poison ratio is ν = 0.45. In the fixed bed (“FB”) experiment, these par- Cases h(m) u* (cm/s) f (kg/m ) f (m /s) d (mm) sρρ= s/ f ws(cm/s) ticles are glued to the bed forming a single layer rough elements, while FB 0.105 5.2 1000 10−6 3 –– the bed is covered by thick layers of particles in the “MB” case, and the MB 0.128 5.0 1000 10−6 3 1.192 5.59 particles are free to move. We first carried out a numerical simulation of the case FB to es- tablish the accuracy of the present numerical model on hydrodynamics condition and sediment properties are summarized in Table 3. The se- before presenting more complicated mobile bed sheet flow model va- 3 diment particles are irregularly shaped with density ρs = 1192 kg/m , lidation. To simulate the flow over fixed rough bottom, a single layer of and median grain diameter d = 3 mm. The resulting mean settling ve- particles are fixed above the bottom boundary in the DEM (i.e., the locity is measured to be ws = 5.59 cm/s. Similar to the case NS1 in particle velocities are zero and their positions are fixed). The rough wall Muste et al. (2005), we used a sphericity of η = 0.594 to match the function (Eq. (34)) is used with a bed roughness ks = 2.5d to estimate settling velocity with the experiment, while the original grain size d is the ωbed in the turbulence closure. In the experiment of Revil-

Fig. 4. The comparison of (a) velocity profile, (b) normalized Reynolds shear stress and (c) TKE profile between numerical results (solid curves) and measured data (filled triangle symbols) for the fixed bed case (‘FB’) in the experiment of Revil- Baudard et al. (2016); In all panels, the fixed particle layer is denoted as circle symbols. In panel (b), the dashed curve is denotes a linear profile of the total shear stress.

443 Z. Cheng et al. Advances in Water Resources 111 (2018) 435–451

Baudard et al. (2016), the ﬂow depth above the ﬁxed particles is about h = 0.105 m. The vertical domain length is chosen to be

Lhdz =+=0.108 m with a uniform grid size of Δz = 0.25 mm. Therefore, the fixed bed layer is resolved by the first 12 grid points above the bottom. The measured bottom frictional velocity in the ex- fl periment is u* = 0.052 m/s. To match the bottom shear stress, the ow driving force is prescribed as fx = 25.8 Pa/m. The model results of the fluid velocity, Reynolds shear stress and the TKE profiles are compared with the measured data in Fig. 4, where the fixed particle layer is also denoted as circle symbols. Due to the drag force from the fixed particles above the bottom, the velocity profile drops to zero within the fixed bed layer, and good agreements in the streamwise velocity profiles are obtained with the measured data. The modeled Reynolds shear stress profile captures the linear decaying shape (dashed curve) and it matches the experimental data reasonably well. In particular, the Reynolds- averaged closure provides a good prediction of the TKE magnitude throughout most of the water column. The good agreement with the FB case confirms that the turbulence closure works well for the steady flow over a rough fixed sediment bed. The mobile bed sheet flow (see case MB in Table 3) is then studied with a thick layer of particles at the bottom of the domain. To prepare the sediment bed, the particle velocities are initialized to be zero, and 43,929 particles are well mixed in the whole domain. Due to the fl fi gravitational settling, the particles settle down to the bottom until their Fig. 5. A snapshot of ow velocity eld (arrows) and sediment particles (assumed to be spherical) for the entire computation domain along with the definition of coordinate kinematic energies are minimized. After this initialization step, the system. The initial bed depth is denote as z , and the water depth is denoted as h. initial bed level locates at z = 0.045 m above the bottom of the domain. b Due to the sediment suspension, the final bed depth at the quasi-steady state will be smaller. Through a preliminary test, we determined that Baudard et al. (2015), the large nearbed shear layer observed in the experiment may be related to the nearbed intermittencies. Even though the total vertical domain height should be Lz = 0.168 m so that the final the EIM is used for the turbulence-sediment interaction, the stochastic flow depth of h = 0.128 m (sediment bed location becomes zb = 0.04 m) can be obtained after the flow reaches the steady state. The vertical model is still too simple to model the bed intermittency, and a turbu- domain is discretized into 168 grid cells with a uniform grid size lence-resolving simulation approach may be necessary for such feature Cheng et al. (2017b). Δz = 0.001 m. The streamwise and spanwise domain lengths are fi L = 0.144 m and L = 0.072 m. In these two horizontal directions, only The sediment concentration pro les normalized by the maximum x y ϕ one CFD grid cell is used in each direction. To confirm the model do- sediment concentration max are compared in Fig. 6b. It shall be noted main size is adequate, we carried out a comparative case by reducing that our numerical model predicts that the maximum sediment concentration is ϕ ≈ 0.635, while the measured data gives ϕ = 0.55. the streamwise domain length by half (Lx = 0.072 m), and the model max max results on mean flow quantities show good convergence. The same The discrepancy in the maximum packing concentration is probably coefficient C = 3 calibrated for dilute suspension (see Section 3.1)is related to the non-spherical particle shape used in the experiments. 0 fi used here for the sheet flow simulation using the LRN k − ω model. The From the normalized sediment concentration pro les, we can see that snapshot of the horizontal fluid velocity profile and sediment particle the modeled sediment concentration with EIM shows a more smooth distribution after the flow reaches the statistically steady state is shown vertical distribution and is more consistent with measured concentra- fi fi in Fig. 5. Although the flow is solved using a Reynolds-averaged tur- tion pro le. On the other hand, the concentration pro le without the bulence closure, the stochastic motions of the sediment particles are EIM indicates that a dense, thin transport layer is predicted between captured by the eddy interaction model and particle collisions. As a 3(dzz<−b )5 < d. Consequently, excessive sediment accumulation fl result of the eddy interaction model, the sediment particles are sus- occurs in this region, and sediment ux is over-predicted (see Fig. 6c). pended away from the bed via turbulent suspension. This feature is similar to typical bedload transport model results for The numerical results of the mean velocity profile, normalized much coarser particles or aeolian transport (Durán et al., 2012). Here, the ‘shoulder-shape’ concentration profile is clearly absent in the concentration profile, sediment fluxes (Qss= ϕu ) and TKE profiles are compared with the measured data in Fig. 6. To reduce the fluctuations measured data and the model result with EIM shows a better agree- due to stochastic motion of particles, time-averaging with a 10 second ment. At the higher Shields parameter and a fall parameter (ratio of window is applied to calculate the mean flow quantity after the flow settling velocity to friction velocity) around 1 or smaller, the suspended reaches steady state. In panel (a), we observe that the modeled fluid transport becomes non-negligible. This point will be discussed more velocity profile is similar to the measured data in the upper water extensively in Section 4.2. Comparisons presented here indicate that the EIM can effectively model the turbulent diffusion of sediment con- column ((zz−>b)7 d) when sediment concentration is very dilute. In the region of intermediate sediment concentration, centration. Therefore, including the eddy-interaction model in Rey- (0 <−()/zzd <7), sediment velocity is slightly smaller than the fluid nolds-averaged formulation is essential to accurately model sediment b fi velocity and agrees with measured velocity profile. This lag in sediment concentration. Although the concentration pro le with C0 = 3 captures phase velocity is consistent with many particle-laden flow observations the main features similar to the measured data, it is clear that the (e.g., Muste et al., 2005; Pal et al., 2016). The modeled velocity profiles present model underpredicts the sediment suspension in the range of fl without the eddy interaction model are similar and hence they are not 5dzz<−()10,b < dand hence the sediment ux is also underpredicted (see panel (c) in Fig. 6). While it is possible to further increase C (in- shown here for brevity. Very near the bed ((zz−≤b)3 d), the model 0 over-predicts the velocity gradient, while the measured data shows a crease turbulent suspension) to match the measured data better, it may fi milder increase of velocity away from the bottom in the range of not be physically valid. The TKE pro les are further compared with the 0 <−()7zz < d. As a result, the numerical model under-predicts the measured data in Fig. 6d. Firstly, we can see that including/excluding b fi shear layer thickness above the bed. According to Revil- the EIM has negligible impact on the modeled TKE pro le, and both

444 Z. Cheng et al. Advances in Water Resources 111 (2018) 435–451

Fig. 6. The comparison of (a) velocity profile, (b) normalized concentration profile and (c) sediment flux profile between numerical results and measured data with eddy interaction model (solid curve) and without eddy interaction model (dashed curve); In all panels, the circle symbols are measured data. In panel (a), solid curve denotes the fluid velocity, and dash- dot curve is the sediment velocity with EIM.

Fig. 7. The comparison of (a) Reynolds shear stress profiles and (b) sediment concentration profiles plotted in semi-log scale for model result (C0 = 2, thick solid curve; C0 = 3, thick magenta dashed curve; C0 = 6, thick black dash-dot curve; C0 = 8, thick blue dash-dot curve) and measured data (symbols). In panel (a), the thin dashed curve denotes a linear fit to the Reynolds shear stress profile. In panel (b), the thin dash-dot curves are the fitted Rouse profile with Rouse number Ro = 4.78, 2.98, 2.42 and 1.64 for the model results of C0 = 2, 3, 6 and 8, respectively. The value for the measured data is Ro = 2.14. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

results show under-prediction of TKE away from the bed through these frequent but intermittent sediment burst events. It is

((zzd−>b)/ 7) and very near the bed ((zzd−

(7(<−zzdb )/25 <) and a signiﬁcant enhancement is also observed under-predicted with C0 = 3 in the EIM, the discrepancy is believed to very near the bed ((zzd−b)/ 7) can be enhanced

445 Z. Cheng et al. Advances in Water Resources 111 (2018) 435–451

Fig. 8. The vertical structure of turbulent eddy viscosity and sediment diﬀusivity are compared in panel (a) and (b), respectively. The corresponding vertical structure of Schmidt number (Sc= νft/ ν p) is plotted in panel (c). In all three panels, model result with C0 = 2 is denoted as thick solid curve, C0 = 3 is denoted as thick dashed curve, C0 = 6 is denoted as thick dash-dot curve and C0 = 8 is denoted as magenta thick dash-dot curve. The symbols are the measured data. In panel (c), the thin dash-dot curves show the mean level of Schmidt number (Sc = 0.44 for measured data, Sc = 1.5, 1, 0.75 and 0.65 for model results with C0 = 2, 3, 6 and 8, respectively).

4. Discussion matches the measured sediment concentration proﬁle, as discussed

before, increasing C0 may not be physically justified because the pre- 4.1. Sensitivity of sediment diffusivity to the coefficient C0 dicted suspended sediment concentration also depends on modeled turbulence quantities. As demonstrated in Section 3.1 for the channel flow with dilute For given sediment properties and flow conditions, the Rouse sediment suspension, the sediment concentration profiles are sensitive number depends on the Schmidt number Sc, which is defined as the to the coefficient C0 in the eddy interaction model, and the suspended ratio between the fluid turbulent eddy viscosity (νft) and the sediment sediment concentration gradient increases with C0 values. It is clear diffusivity (νp). In many Reynolds-averaged Eulerian simulations of that the gradient of sediment concentration profile is related to the sediment transport (e.g., Hsu et al., 2004; Revil-Baudard and Chauchat, particle dispersion (or sediment diffusion). In this section, we further 2013; Cheng et al., 2017a), the gradient transport assumption is analyze the sensitivity of the suspended sediment concentrations and adopted, the sediment diffusivity to the coefficient C0 under steady sheet flow ′ ∂ϕ conditions by varying C0= 2, 3, 6, and 8. ws ϕν′ =− , p (40) The effect of C0 values on the sediment concentration profile is il- ∂z lustrated in Fig. 7. Similar to the Rouse profile in dilute particle-laden where the sediment diffusivity is often parameterized by the turbulent flows (Eq. (38)), the Rouse profile in the sheet flow can be determined eddy viscosity, ννSc= /,with a constant Schmidt number (e.g., Hsu as, pft et al., 2003; Chen et al., 2011; Cheng et al., 2017a). Alternatively, the −Ro sediment diffusivity may be evaluated as νwϕϕzps=−/( ∂ / ∂ ) by con- ϕ ⎛ zz− b hz+−br z⎞ = ⎜⎟, sidering the balance between the turbulent suspension flux and the ϕr hz+−b zzzrb− (39) ⎝ ⎠ s′ settling flux, w ϕwϕ′ = s . In the present model, the sediment motion is In practice, the Rouse profile is only applicable when the turbulent directly resolved by a Lagrangian approach, and the eddy interaction suspension is dominant while the particle–particle interactions are model is incorporated to simulate the sediment suspension by the flow negligible. Therefore, the reference location zr is chosen to be the turbulence. Therefore, it is interesting to evaluate the eddy interaction lowest elevation at which the Reynolds shear stress follows a linear model in terms of the resulting sediment diffusivity and Schmidt profile. The shear stress profiles corresponding to different C0 values are number. shown in Fig. 7a. The Reynolds shear stress profile is nearly unaffected The vertical profiles of turbulent eddy viscosity and sediment dif- by the C0 value. Meanwhile, the Reynolds shear stress follows the linear fusivity for C0=2, 3, 6 and 8 are compared in Fig. 8(a) and (b). The distribution above (zzd−=b)/ 7.5, and therefore it can be conjectured turbulent eddy viscosity profiles obtained using different C0 values are that the inter-granular stress becomes important below (zzd−=b)/ 7.5 similar to each other and their vertical distributions are close to the and a common reference location zzrb=+7.5 dis chosen. measured data. However, the magnitude of the eddy viscosity is over- The normalized sediment concentration profiles are plotted in predicted compared with the measured data. Recall in Fig. 6(d) that the logarithmic scale in Fig. 7b, where the thick curves are numerical represent model also underpredicts TKE, we can conclude that the model sults, and thin dash-dot curves are the corresponding fitted Rouse may significantly underpredict ω due to inability to resolve intermittent profiles. The modeled sediment concentration profiles fit the Rouse turbulent motion and sediment burst. This may provide some useful profile well in the dilute region ((zz−>b)7 d) for all the C0 values insights to further improve the present k − ω model for two-phase flow tested. However, different slopes of concentration profiles were ob- in the future. As shown in Fig. 8(b), the vertical profiles of the sediment served by varying C0 values. We quantify the slope of sediment con- diffusivities are sensitive to the C0 values (see Fig. 8(b)), and the se- centration in logarithmic scale using the Rouse number Ro (see diment diffusivity increases with the increasing values of C0. Because Eq. (39)). For C0 = 2, nearly no sediment is suspended above discrepancies exist in both eddy viscosity and sediment diffusivity, the (zzd−=b)/ 15 and the Rouse number Ro = 4.78 is large compared with overall evaluation was also examined by the ratio of these two quan- the measured data Ro = 2.14. Using C0 = 3, sediments are suspended tities, namely the Schmidt number. The resulting Schmidt numbers much higher in the water column and the resulting Ro = 2.98 is sig- (Sc= νft/ ν p) are presented in Fig. 8(c). We noticed that the predicted nificantly lower. Further increasing C0 to 6 and 8, the Rouse number Schmidt number was more or less a constant in the suspension layer reduced to 2.42 and 1.64. Although the model result using C0 = 6 (zz−>b)6 dfor all the runs regardless of C0 values, and this feature is

446 Z. Cheng et al. Advances in Water Resources 111 (2018) 435–451

consistent with the measurement. With C0 = 3 the resulting Schmidt dilation causes a larger erosion depth in the dense layer (ϕϕ/0.5max > number is around unity (Sc ≈ 1), which is significantly larger than the or (zzd−b)/ 8 for the case with the Reynolds-averaged model to capture the nearbed intermittencies as highest Shields parameter). On the other hand, when EIM is in- observed in the sheet flow experiment of Revil-Baudard et al. (2015). corporated to model turbulent suspension, sediments are suspended The nearbed intermittency enhances the turbulent intensities within the further away from the bed. The sediment transport flux in the relatively fi fl dense layer and upper water column. As a result, the present model dilute layer (ϕϕ/0.3max < ) is signi cantly larger, and the total ux is underpredicted turbulent intensity in these regions, which can further expected to be larger compared with the cases without EIM (see cause the underprediction of the suspended sediment concentration. To Fig. 9b). fully understand the dependence of Schmidt number on turbulent flow The sediment transport rate can be obtained by integrating the se- fl characteristics and sediment properties, a more sophisticated turbu- diment transport ux (Qs) over the entire vertical domain, and the di- lence-resolving models may be needed. Secondly, several interphase mensionless sediment transport rate can be computed as (Durán et al., momentum transfer forces such as the added mass and lift forces are 2012), neglected in the present study. It is expected that these interphase N ∑ viiVLL/( x y ) transfer forces are less important for heavy sand particles. However, Ψ = i=1 . they can become important for lightweight coarse particles (Jha and (1)sgd− 3 (41) Bombardelli, 2010). Finally, we shall note that detailed experimental measurements on natural sand transport in sheet flow are needed to In this study, the sediment transport rate obtained with EIM is denoted Ψ Ψ study the relevance of this nearbed intermittency of lightweight parti- as t, while the transport rate without EIM is denoted as b. According cles for the sand transport. More comprehensive investigations are to the previous experimental results on the sediment transport rate, a warranted for future work. general form of power law relationships between the dimensionless sediment transport rate and the excess Shields parameter (θθ− c) can be written as,

4.2. Transport rate and transport layer thickness N Ψ(),=−Mθ0 θc 0 (42)

The present model is applied to study the role of turbulent sus- Where a typical critical Shields parameter θc = 0.05 is used, several pension (modeled by EIM) on sediment transport rate and transport different values of the coefficient M0 and N0 were proposed from var- layer thickness. In sediment transport applications, the sediment ious experimental results. On the basis of the flume experiments for transport rate is often of high interest, as it is directly used in regional- rather coarse sand (d > 3 mm) at low Shields parameter (θ < 0.2), scale models to study morphological evolutions (e.g., Lesser et al., 2004; Meyer-Peter and Muller (1948) proposed that M0 = 5.7 and N0 = 1.5. Warner et al., 2008). Many steady flow experiments revealed that the This is the well-known power law where the transport rate is propor- dimensionless sediment transport rate can be parameterized by the non- tional to the 3/2 power of the excess Shields parameter ()θθ− c . Based dimensional bottom shear stress (e.g., Meyer-Peter and Muller, 1948; on the duct flow experiment with a smaller grain size (d = 0.7 mm) at Nnadi and Wilson, 1992; Ribberink, 1998). The non-dimensional form higher Shields parameters (θ > 1), Nnadi and Wilson (1992) suggested of the bottom shear stress is called Shields parameter, that the coefficient M0 should be increased to M0 = 12. More recent θτρ=−b/[(sf ρgd ) ]. To evaluate the model capability to predict sedi- study by Ribberink (1998) found that the power 3/2 should be in- ment transport rate, we carried out 14 cases with Shields parameter creased to about 1.67 as the suspended load becomes important when ranging from θ = 0.3 to 1.2 with/without EIM (see Table 4). the Shields parameter becomes larger. The resulting sediment concentration profiles and sediment flux The numerical results of the dimensionless sediment transport rates profiles for three representative Shields parameters (θ=0.5, 0.8 and 1.2) as a function of the Shields parameters are plotted in Fig. 10. Clearly, are shown in Fig. 9, where panels (a, b) corresponds to the results with the sediment transport rates predicted with EIM (circle symbols) and EIM, and panels (c, d) corresponds to the results without EIM. As the without EIM (triangle symbols) increase rapidly when the Shields shear stress exerted on the granular bed increases, the shear-induced parameter increases, and this trend follows the empirical power law (Eq. (42)) very well. The dash-dot curve in Fig. 10 shows the power law

Table 4 with N0 = 1.5 (Meyer-Peter and Muller, 1948) and the resulting best fit A summary of the numerical experiments to study the effect of EIM on the sediment is M0 = 8.1. However, the fitted curve with a power of N0 = 1.5 over- transport rate and transport layer thickness at various Shields parameters. The transport rate and transport layer thickness with EIM are denoted as Ψ and δ , respectively, while predicts the sediment transport rate for lower Shields parameters t t θ the results without EIM are denoted as Ψb and δb, respectively. ( < 1), while the transport rate in the higher Shields parameter range is under-predicted. On the other hand, the best fit of the power law for d (mm) u (cm/s) θ Fwu= / Ψ Ψ δ /d δ /d * s * b t b t the present model results gives M0 = 8.27 and N0 = 2.0, which is consistent the values reported by Ribberink (1998), M = 10.4 and 3 3.87 0.3 1.44 0.48 0.67 2.28 2.90 0 3 4.47 0.4 1.25 0.93 1.16 2.90 4.15 N0 = 1.67. In addition, the transport rate without EIM is also compared 3 5.0 0.5 1.12 1.40 1.89 3.32 5.39 with that of EIM. It is evident that the transport rate without EIM is 3 5.48 0.6 1.02 1.81 2.85 3.94 6.43 generally smaller, and the discrepancy increases as the Shields para- 3 6.32 0.8 0.88 3.03 4.20 4.98 7.88 meter increases. If we further fit the transport rate obtained without 3 7.07 1.0 0.79 5.01 7.39 6.01 10.16 3 7.74 1.2 0.72 7.67 11.05 7.05 12.44 EIM into the power law formula, we obtain that M0 = 5.5 and N0 = 2.0. It is interesting to note that although the proportionality constant M0 is

447 Z. Cheng et al. Advances in Water Resources 111 (2018) 435–451

Fig. 9. The sediment concentration profile and transport flux profile at three different Shields parameter, θ = 0.5 (solid curve), θ = 0.8 (dashed curve) and θ = 1.2 (dash-dot curve). Panel (a) and (b) corresponds to the result with eddy interaction model, while panel (c) and (d) are the results without eddy interaction model. The sediment concentration is normalized by the maximum sediment concentration fl ϕmax = 0.635, and the transport ux is normalized by (1)sgd− .

much lower than that of EIM, the power N0 remains the same. As shown in Fig. 9 (b) and (d), the sediment horizontal flux mainly occurs within a thick layer of about 10 ∼ 15 grain diameters above the bed. In sheet flow applications, the transport layer thickness is another quantity of interest, because this is where a large portion of transport takes place. For example, Wilson (1987) argues that mobile beds at high shear stresses can neither be considered as a rough or smooth fixed wall but they obey their own friction law with a frictional length scale proportional to the thickness of the major transport layer. Wilson (1987) defined the major transport layer thickness as the dis-

tance of the lowest mobile bed layer (us < 1 mm/s) and the sediment concentration ϕ = 8%. However, we noticed that using the 8% threshold may neglect too much transport for the present analysis and a lower threshold may be more appropriate. Here, we define the transport layer directly from the sediment flux profile, where the dimensionless sediment flux is larger than a small threshold: Qs/ (sgd−> 1) 0.05. The resulting transport layer thickness with EIM (δt) and without EIM (δb) are compared in Fig. 10b. It is evident that the transport layer thickness increases with the Shields parameter. Ac- cording to the experimental observations (e.g., Wilson, 1987; Sumer et al., 1996), the transport layer thickness is nearly proportional to the grain diameter and Shields parameter. As shown in Fig. 10b, we can see that a linear relationship can be found regardless of whether EIM is Fig. 10. The nondimensional transport rate (panel a) and transport layer thickness (panel adopted or not, even though the proportionality coefficients are quite b) as a function of Shields parameter θ. The circled symbols are model results with eddy different. Without EIM, the transport layer thickness can be well de- interaction model (denoted as Ψ ). To contrast the effect of EIM, the model results without t scribed as δdb/6.18= θ. However, the transport layer thickness with EIM (Ψb) are denoted as triangle symbols. The solid curve shows the empirical formula- EIM is much larger, δds/= 10.28θ with the proportional coefficient very tion of Eq. (42) with θ = 0.05, M = 8.27 and N = 1.97, while the dash-dot curve cor- c 0 0 close to the value 10 as suggested by Wilson (1987). responds to θc = 0.05, M0 = 8.1 and N0 = 1.5. The best fit to the transport rate without According to Bagnold (1966), the particle suspension occurs when EIM is Eq. (42) with θc = 0.05, M0 = 5.5 and N0 = 2.0. In panel (b), the solid curve is the linear fit transport layer thickness with EIM, while the dashed curve is for the cases the dominant vertical velocity of the turbulent eddies exceeds the without EIM. particle settling velocity. Assuming that the vertical velocity fluctuation

448 Z. Cheng et al. Advances in Water Resources 111 (2018) 435–451 can be quantified by the vertical turbulent velocity fluctuation, we can 5. Conclusion assume that the turbulent suspension is important if wrms′ > ws. In the present model, an isotropic turbulence is assumed, such that the ver- In this paper, a Reynolds-averaged Euler–Lagrange sediment trans- tical turbulence intensity is approximated as, wrms′ ≈ 2/3k . port model was developed and applied to steady sheet flow, where the Nezu (1993) suggested that the maximum TKE can be estimated as inter-granular interaction is directly resolved and the turbulent sus- 4.78u 2 for turbulent flow over smooth bed. In the present sheet sedi- * pension of particles is modeled using an eddy interaction model. A LRN ment transport with coarse light particles, the maximum TKE can be k − ω model extended for two-phase flow is implemented for the flow 2 reasonably represented by 3u (see Fig. 6d), thus the turbulent sus- turbulence, which also provides the required turbulence statistics for * fi pension can be initiated when the shear velocity satis es, ws/u* < 2 . the eddy-interaction model. The eddy interaction model was first cali- This is similar to the discussion of van Rijn (1984b) and brated using the dilute suspension experiments of Kiger and Pan (2002) Sumer et al. (1996), where they suggested that the relative importance and Muste et al. (2005). While the model is able to predict the mea- between suspended load and bedload sediment transport can be cate- sured flow velocity and turbulence kinetic energy very well, the model gorized by the fall parameter, Fwu= / . As summarized in Table 4, the ffi s * results are found to be sensitive to the coe cient C0 associated with the fall parameter varies from 1.44 to 0.72 as the shear velocity increases eddy-particle interaction time (see Eq. (36)), and a value of C0 ≈ 3is from 3.87 to 7.74 cm/s. From the previous discussion on the sediment calibrated to match the measured concentration profile in the dilute ff transport rate and transport layer thickness, it is found that the di er- particle-laden flow. ence of the transport rate between the results with and without EIM is After calibrating the eddy-interaction model for dilute suspension, negligible when the Shields parameter is smaller than 0.5 (fall para- an application of CFDEM-EIM to steady sheet flow was carried out by ≥ meter F 1.25). However, when the Shields parameter is larger than simulating the laboratory experiment of Revil-Baudard et al. (2015) 0.5 (or F < 1.25), the difference becomes noticeable. with C0 = 3. Although good agreements for flow velocity, turbulence To carry out more quantitative analysis, we consider that the tur- kinetic energy, sediment concentration and sediment flux profiles are fi bulent suspension is most signi cant for the suspended load, which obtained for most of the sheet flow layer, the model clearly under- ϕ mainly occurs in the region of < 0.08. The non-dimensional sus- predicts turbulence and suspended sediment concentration in the dilute fi pended load sediment transport rate can be de ned as, region. The underpredicted suspended sediment concentration is quantified by sediment diffusivity and we found that the sediment 1 Lz ff ffi fl Ψsus = ∫ ϕus dz. di usivity decreases as the coe cient C0 increases, while the uid (1)sgd− 3 zϕ(= 0.08) (43) turbulent eddy viscosity is not sensitive to C0 values. As a result, the resulting Schmidt number (ratio of fluid eddy viscosity to the sediment To illustrate the importance of EIM on the prediction of suspended diffusivity) reduces as C increases. However, the Schmidt number fl 0 sediment ux, the suspended load with/without EIM are compared in cannot be reduced to the measured value of 0.44 unless an unrealistic Fig. 11 for a range of fall parameters. As the fall parameter increases, large value of C0 is used. Therefore, it is likely that the under-prediction the sediment particles are less likely to be suspended by the turbulent of suspended sediment concentration in the dilute region is mainly due eddies, thus the suspended sediment transport reduces rapidly. This is to underprediction of turbulence kinetic energy above the major sheet fi con rmed by the results of EIM, where the non-dimensional suspended flow layer. As the higher level of turbulence may be associated with sediment transport rate is reduced from 1.4 to 0.2 when the fall para- intermittent sediment burst events especially pronounced for light- meter increases from 0.72 to 1.44. However, we can see that the sus- weight particles (Revil-Baudard et al., 2015), a turbulence-resolving pended load predicted without EIM is quite small (around 0.2) and approach for the present Euler–Lagrange model may be necessary. more or less a constant independent of the fall parameter. This indicates Meanwhile, as the model can reproduce the major features of sheet flow fl that the EIM is essential to capture the suspended sediment ux. For layer, a model investigation was carried out to investigate the role of fl fi F < 1.25, suspended load ux can be signi cantly under-predicted and EIM and the resulting turbulent suspension on sediment transport rate EIM should be included in the Euler-Lagrange model for steady sheet and transport layer thickness. Model results confirmed that the non- fl ows. dimensional transport rate follows a power law with the Shields parameter consistent with empirical formulations. Significant underprediction of sediment transport rate were obtained without EIM due to lack of turbulent suspension, and the discrepancy between the result of EIM and without EIM is more pronounced when the fall parameter is lower than 1.25 (relatively smaller setting velocity or larger bottom friction velocity). Further analysis on transport layer thickness suggests that only when EIM is incorporated, the model is able to reproduce the well-known formula suggested by Wilson (1987). Future improvements of the present CFDEM-EIM are suggested in the following aspects: First, the eddy interaction model is included only in the drag force, while the other interphase momentum transfer forces such as added mass and lift forces are ignored. However, their relative importance to the drag force in the eddy interaction model needs more investigations, especially for lightweight coarse particles. Secondly, even though the particles are tracked in a 3D domain with a Lagrangian approach, the fluid is solved only in a 1DV domain, and the flow is assumed to be homogeneous in the streamwise and spanwise directions. This assumption is reasonable for typical sheet flow conditions. However, for flows over nonuniform bathymetry or bedforms, this assumption is violated, and multi-dimensional simulations are needed for fl Fig. 11. Nondimensional suspended sediment transport rate Ψsus in the dilute region the uid phase. Thirdly, the turbulence is assumed to be homogeneous ϕ ( < 0.08) as a function of the fall parameter Fwu= s/ *. The circled symbols are model and isotropic, thus the eddy interaction model may be too simple to results with eddy interaction model, while the triangle symbol denotes the transport rate reproduce the inhomogeneous features such as turbulent burst and obtained without EIM. preferential concentrations. Within the context of turbulence-averaged

449 Z. Cheng et al. Advances in Water Resources 111 (2018) 435–451 formulation, more sophisticated turbulence closure and eddy-interac- dx.doi.org/10.1029/2007JC004237. tion schemes can be pursued. Fourthly, it is noted that the model results Capecelatro, J., Desjardins, O., 2013. An Euler–Lagrange strategy for simulating particle–laden flows. J. Comput. Phys. 238, 1–31. ISSN 0021-9991. are sensitive to the estimation of eddy life time, which is also highly Chauchat, J., Cheng, Z., Nagel, T., Bonamy, C., Hsu, T.-J., 2017. Sedfoam-2.0: a 3d two- variable based on the flow condition (Coimbra et al., 1998), and a more phase flow numerical model for sediment transport. Geosci. Model Dev. Discuss. sophisticated turbulence model that directly resolves the eddy life time 2017, 1–42. ISSN 1991-962X. Chen, X., Li, Y., Niu, X., Chen, D., Yu, X., 2011. A two-phase approach to wave-induced will be highly viable. Furthermore, to make good use of the coupled sediment transport under sheet flow conditions. Coastal Eng. 58 (11), 1072–1088. Euler-Lagrange scheme, CFDEM-EIM should be extensively applied to ISSN 0378-3839. study the effects of grain size distribution and grain shape on sediment Cheng, Z., Hsu, T.-J., Calantoni, J., 2017a. Sedfoam: a multi-dimensional Eulerian two- transport (Calantoni et al., 2004; Calantoni and Thaxton, 2008; phase model for sediment transport and its application to momentary bed failure. Coastal Eng. 119, 32–50. ISSN 0378-3839. Fukuoka et al., 2014; Harada and Gotoh, 2008; Harada et al., 2015). Cheng, Z., Hsu, T.-J., Chauchat, J. 2017b. An Eulerian two-phase model for steady sheet Finally, the present study focused on developing a robust turbulence- flow using large-eddy simulation methodology. Adv. Water Resour. in press. – Coimbra, C., Shirolkar, J., McQuay, M.Q., 1998. Modeling particle dispersion in a tur- averaged Euler Lagrange model for various sediment transport appli- – fi bulent, multiphase mixing layer. J. Wind Eng. Ind. Aerodyn. 73 (1), 79 97. ISSN cations. However, we also identi ed several outstanding issues in sheet 0167-6105. flow sediment transport requiring further investigations, such as near Cundall, P.A., Strack, O.D., 1979. A discrete numerical model for granular assemblies. – bed intermittency and sediment diffusivity, which may require a tur- Geotechnique 29 (1), 47 65. ISSN 0016-8505. Dong, P., Zhang, K., 2002. Intense near-bed sediment motions in waves and currents. bulence-resolving simulation approach. Clearly, a fundamental under- Coastal Eng. 45 (2), 75–87. ISSN 0378-3839. standing on many aspects of turbulence-particle interactions must be Drake, T.G., Calantoni, J., 2001. Discrete particle model for sheet flow sediment transport addressed by turbulence-resolving simulations and some encouraging in the nearshore. J. Geophys. Res.: Oceans 106 (C9), 19859–19868. ISSN 2156-2202. Durán, O., Andreotti, B., Claudin, P., 2012. Numerical simulation of turbulent sediment works using the CFDEM framework have been reported (Schmeeckle, transport, from bed load to saltation. Phys. Fluids 24 (10), 103306. ISSN 1070-6631. 2014; Sun and Xiao, 2016b). Felice, R.D., 1994. The voidage function for fluid-particle interaction systems. Int. J. Multiphase Flow 20 (1), 153–159. ISSN 0301-9322. Finn, J.R., Li, M., Apte, S.V., 2016. Particle based modelling and simulation of natural Acknowledgement sand dynamics in the wave bottom boundary layer. J. Fluid Mech. 796, 340–385. ISSN 0022–1120. Z. Cheng and T.-J. Hsu were supported by the U.S. Office of Naval Fuhrman, D.R., Dixen, M., Jacobsen, N.G., 2010. Physically-consistent wall boundary – Research (N00014-16-1-2853) and National Science Foundation (OCE- conditions for the k- turbulence model. J. Hydraul. Res. 48 (6), 793 800. ISSN 0022–1686. 1537231). J. Chauchat was supported by the Région Rhones-Alpes Fukuoka, S., Fukuda, T., Uchida, T., 2014. Effects of sizes and shapes of gravel particles on (COOPERA project and Explora Pro grant) and the French national sediment transports and bed variations in a numerical movable-bed channel. Adv. Water Resour. 72, 84–96. ISSN 0309-1708. programme EC2CO-LEFE MODSED. J. Calantoni was supported under fl ffi Goniva, C., Kloss, C., Deen, N.G., Kuipers, J.A., Pirker, S., 2012. In uence of rolling base funding to the U.S. Naval Research Laboratory from the U.S. O ce friction on single spout fluidized bed simulation. Particuology 10 (5), 582–591. ISSN of Naval Research. The authors would also like to acknowledge the 1674-2001. “ Gosman, A., Loannides, E., 1983. Aspects of computer simulation of liquid-fueled com- support from the program on Fluid-Mediated Particle Transport in – ” bustors. J. Energy 7 (6), 482 490. ISSN 0146-0412. Geophysical Flows at the Kavli Institute for Theoretical Physics, Santa Graham, D., 1996. On the inertia effect in eddy interaction models. Int. J. Multiphase Barbara, USA. The laboratory LEGI is part of the LabEx Tec 21 Flow 22 (1), 177–184. ISSN 0301-9322. (Investissements d’Avenir - grant agreement nANR-11-LABX-0030) and Graham, D., James, P., 1996. Turbulent dispersion of particles using eddy interaction models. Int. J. Multiphase Flow 22 (1), 157–175. ISSN 0301-9322. Labex OSUG@2020 (ANR10 LABX56). Numerical simulations were Guizien, K., Dohmen-Janssen, M., Vittori, G., 2003. 1DV bottom boundary layer modeling carried out on MILLS/FARBER at the University of Delaware. under combined wave and current: Turbulent separation and phase lag effects. J. Simulations also leverage computing resource provided by Extreme Geophys. Res. 108 (C1), 3016. http://dx.doi.org/10.1029/2001JC001292. Haider, A., Levenspiel, O., 1989. Drag coefficient and terminal velocity of spherical and Science and Engineering Discovery Environment (XSEDE) (TG- nonspherical particles. Powder Technol. 58 (1), 63–70. ISSN 0032-5910. OCE100015). We are grateful to the developers involved in Harada, E., Gotoh, H., 2008. Computational mechanics of vertical sorting of sediment in OpenFOAM, LIGGGHTS, LAMMPS and CFDEM, which are the founda- sheetflow regime by 3d granular material model. Coastal Eng. J. 50 (01), 19–45. ISSN tion of the CFDEM-EIM presented in this paper. Dr. Xiao Yu, Mr. Samuel 0578-5634. Harada, E., Gotoh, H., Tsuruta, N., 2015. Vertical sorting process under oscillatory sheet Bateman and Dr. Raphael Maurin provided many useful comments and flow condition by resolved discrete particle model. J. Hydraul. Res. 53 (3), 332–350. insights to these open source codes and the EIM model. ISSN 0022-1686. Hsu, T.-J., Jenkins, J.T., Liu, P.L.-F., 2003. On two-phase sediment transport: Dilute flow. J. Geophys. Res. 108 (C3), 3057. http://dx.doi.org/10.1029/2001JC001276. References Hsu, T.-J., Jenkins, J.T., Liu, P.L.-F., 2004. On two-phase sediment transport: sheet flow of massive particles. Proc. R. Soc. London, Ser. A: Math. Phys. Eng. Sci. 460 (2048), – Allison, M.A., Demas, C.R., Ebersole, B.A., Kleiss, B.A., Little, C.D., Meselhe, E.A., Powell, 2223 2250. ISSN 1364-5021. fl N.J., Pratt, T.C., Vosburg, B.M., 2012. A water and sediment budget for the lower Jenkins, J.T., Hanes, D.M., 1998. Collisional sheet ows of sediment driven by a turbulent fl – Mississippi-Atchafalaya River in flood years 2008 ∼ 2010: implications for sediment uid. J. Fluid Mech. 370, 29 52. ISSN 1469-7645. fl discharge to the oceans and coastal restoration in Louisiana. J. Hydrol. (Amst.) 432, Jha, S.K., Bombardelli, F.A., 2010. Toward two-phase ow modeling of nondilute sedi- 84–97. ISSN 0022-1694. ment transport in open channels. J. Geophys. Res.: Earth Surf. 115 (F03015). ISSN Amoudry, L., Hsu, T.-J., Liu, P.L.-F., 2008. Two-phase model for sand transport in sheet 2156-2202. flow regime. J. Geophys. Res. 113, C03011. http://dx.doi.org/10.1029/ Jha, S.K., Bombardelli, F.A., 2009. Two-phase modeling of turbulence in dilute sediment- fl 2007JC004179. laden, open-channel ows. Environ. Fluid Mech. 9 (2), 237. Amoudry, L.O., 2014. Extension of kω− turbulence closure to two-phase sediment Kallio, G., Reeks, M., 1989. A numerical simulation of particle deposition in turbulent – – transport modelling: application to oscillatory sheet flows. Adv. Water Resour. 72, boundary layers. Int. J. Multiphase Flow 15 (3), 433 446. ISSN 0301 9322. fi 110–121. ISSN 0309-1708. Keshtpoor, M., Puleo, J.A., Gebert, J., Plant, N.G., 2013. Beach response to a xed sand – – Bagnold, R., 1966. An approach to the sediment transport problem from general physics. bypassing system. Coastal Eng. 73, 28 42. ISSN 0378 3839. fi ff Tech. rep. US Geological Survey Professional Paper. Kiger, K., Pan, C., 2002. Suspension and turbulence modi cation e ects of solid parti- fl – Balachandar, S., Eaton, J.K., 2010. Turbulent dispersed multiphase flow. Annu. Rev. Fluid culates on a horizontal turbulent channel ow. J. Turbul. 3 (19), 1 17. Mech. 42, 111–133. ISSN 0066-4189. Kloss, C., Goniva, C., Hager, A., Amberger, S., Pirker, S., 2012. Models, algorithms and Boyer, F., Guazzelli, E., Pouliquen, O., 2011. Unifying suspension and granular rheology. validation for open source DEM and CFDDEM. Prog. Comput. Fluid Dyn. Int. J. 12 – – Phys. Rev. Lett. 107 (18), 188301. (2 3), 140 152. ISSN 1468-4349. Bredberg, J., Peng, S., Davidson, L., 2000. On the wall boundary condition for computing Lesser, G., Roelvink, J., van Kester, J., Stelling, G., 2004. Development and validation of a – turbulent heat transfer with kω− models. ASME-Publications-HTD. 366. pp. three-dimensional morphological model. Coastal Eng. 51 (89), 883 915. ISSN 0378- 243–250. ISSN 0272-5673. 3839. fl fl Calantoni, J., Holland, K.T., Drake, T.G., 2004. Modelling sheet-flow sediment transport Liu, H., Sato, S., 2006. A two-phase ow model for asymmetric sheet ow conditions. – in wave-bottom boundary layers using discrete-element modelling. Philos. Trans.–R. Coastal Eng. 53 (10), 825 843. ISSN 0378-3839. fl fl Soc. London, Ser. A: Math. Phys. Eng. Sci. 362, 1987–2002. ISSN 1364-503X. Longo, S., 2005. Two-phase ow modeling of sediment motion in sheet- ows above plane – Calantoni, J., Thaxton, C.S., 2008. Simple power law for transport ratio with bimodal beds. J. Hydraul. Eng. 131 (5), 366 379. ISSN 0733-9429. distributions of coarse sediments under waves. J. Geophys. Res. 113, C03003. http:// Luding, S., 2008. Introduction to discrete element methods: basic of contact force models and how to perform the micro-macro transition to continuum theory. Eur. J. Environ.

450 Z. Cheng et al. Advances in Water Resources 111 (2018) 435–451

Civil Eng. 12 (7–8), 785–826. ISSN 1964-8189. induced by the bed mobility in intense sediment-laden flows. J. Fluid Mech. 808, Matida, E., Finlay, W., Lange, C., Grgic, B., 2004. Improved numerical simulation of 469–484. ISSN 0022-1120. aerosol deposition in an idealized mouth-throat. J. Aerosol. Sci. 35 (1), 1–19. ISSN Ribberink, J.S., 1998. Bed-load transport for steady flows and unsteady oscillatory flows. 0021–8502. Coastal Eng. 34 (12), 59–82. ISSN 0378-3839. Maurin, R., Chauchat, J., Chareyre, B., Frey, P., 2015. A minimal coupled fluid-discrete van Rijn, L., 1984a. Sediment transport, part I: bed load transport. J. Hydraul. Eng. 110 element model for bedload transport. Phys. Fluids (1994-present) 27 (11), 113302. (10), 1431–1456. ISSN 0733-9429. ISSN 1070-6631. van Rijn, L., 1984b. Sediment transport, part II: suspended load transport. J. Hydraul. Maxey, M.R., Riley, J.J., 1983. Equation of motion for a small rigid sphere in a nonuni- Eng. 110 (11), 1613–1641. ISSN 0733–9429. form flow. Phys. Fluids (1958–1988) 26 (4), 883–889. Rodi, W., 1987. Examples of calculation methods for flow and mixing in stratified fluids. Mehrotra, V., Silcox, G.D., Smith, P.J., 1998. In: Anon (Ed.), Numerical Simulation of J. Geophys. Res. 92 (C5), 5305–5328. ISSN 2156-2202. Turbulent Particle Dispersion Using a Monte Carlo Approach. American Society of Rusche, H., 2002. Computational fluid dynamics of dispersed two-phase flows at high Mechanical Engineers, Fluids Engineering Division (Publication) FED, ASME, phase fractions. (Ph.D. thesis). Imperial College London (University of London). Fairfield, NJ, United States. Schmeeckle, M.W., 2014. Numerical simulation of turbulence and sediment transport of Menter, F., Esch, T., 2001. Elements of industrial heat transfer predictions. In: 16th medium sand. J. Geophys. Res.: Earth Surf. 119 (6), 1240–1262. ISSN 2169-9011. Brazilian Congress of Mechanical Engineering (COBEM). Vol. 109. Shi, H., Yu, X., 2015. An effective Euler-Lagrange model for suspended sediment transport Meyer-Peter, E., Muller, R., 1948. Formulas for bed-load transport. In: Hydraulic by open channel flows. Int. J. Sediment Res. 30 (4), 361–370. ISSN 1001-6279. Engineering Reports. IAHSR 2nd meeting, Stockholm, Appendix 2. Simeonov, J.A., Bateman, S., Calantoni, J., 2015. Filter width and uncertainty estimation Milojeviè, D., 1990. Lagrangian stochastic-deterministic (LSD) predictions of particle in continuum modeling of particle phases. Int. J. Multiphase Flow 74, 79–83. ISSN dispersion in turbulence. Part. Part. Syst. Charact. 7 (1–4), 181–190. ISSN 0301-9322. 1521–4117. Sumer, B.M., Kozakiewicz, A., Fredsoe, J., Deigaard, R., 1996. Velocity and concentration Mindlin, R., 1949. Compliance of elastic bodies in contact. J. Appl. Mech. 16, 259–268. profiles in sheet-flow layer of movable bed. J. Hydraul. Eng. 122 (10), 549–558. ISSN Moser, R.D., Kim, J., Mansour, N.N., 1999. Direct numerical simulation of turbulent 0733-9429. channel flow up to Re = 590. Phys. Fluids 11 (4), 943–945. ISSN 1070-6631. Sun, R., Xiao, H., 2016a. Sedifoam: a general-purpose, open-source CFDDEM solver for Mossa, J., 1996. Sediment dynamics in the lowermost Mississippi River. Eng. Geol. 45 (1), particle-laden flow with emphasis on sediment transport. Comput. Geosci. 89, 457–479. ISSN 0013-7952. 207–219. ISSN 0098-3004. Muste, M., Yu, K., Fujita, I., Ettema, R., 2005. Two-phase versus mixed-flow perspective on Sun, R., Xiao, H., 2016b. CFD-DEM Simulations of current-induced dune formation and suspended sediment transport in turbulent channel flows. Water Resour. Res. 41, morphological evolution. Adv. Water Resour. 92, 228–239. ISSN 0309-1708. W10402. http://dx.doi.org/10.1029/2004WR003595. Sweby, P.K., 1984. High resolution schemes using flux limiters for hyperbolic conserva- Nezu, I., 1993. Turbulence in Open-Channel Flows. A. A. Balkema. tion laws. SIAM J. Numer. Anal. 21 (5), 995–1011. ISSN 00361429. Nnadi, F.N., Wilson, K.C., 1992. Motion of contact-load particles at high shear stress. J. Taylor, G.I., 1922. Diffusion by continuous movements. Proc. London Math. Soc. 20 (1), Hydraul. Eng. 118 (12), 1670–1684. 196–212. O’Donoghue, T., Wright, S., 2004. Concentrations in oscillatory sheet flow for well sorted Vanoni, V., 2006. Sedimentation Engineering. American Society of Civil Engineers. and graded sands. Coastal Eng. 50 (3), 117–138. ISSN 0378–3839. Warner, J.C., Sherwood, C.R., Signell, R.P., Harris, C.K., Arango, H.G., 2008. Pal, D., Jha, S.K., Ghoshal, K., 2016. Velocity lag between particle and liquid in sediment- Development of a three-dimensional, regional, coupled wave, current, and sediment- laden open channel turbulent flow. Eur. J. Mech. B. Fluids 56, 130–142. ISSN 0997- transport model. Comput. Geosci. 34 (10), 1284–1306. ISSN 0098-3004. 7546. Weller, H., 2002. Derivation, modelling and solution of the conditionally averaged two- Pirker, S., Kahrimanovic, D., Goniva, C., 2011. Improving the applicability of discrete phase flow equations. Tech. rep. OpenCFD Ltd. phase simulations by smoothening their exchange fields. Appl. Math. Model. 35 (5), Wilcox, D.C., 1988. Reassessment of the scale-determining equation for advanced tur- 2479 –2488. ISSN 0307-904X. bulence models. AIAA J. 26 (11), 1299–1310. ISSN 0001-1452. Renzo, A.D., Maio, F.P.D., 2005. An improved integral non-linear model for the contact of Wilcox, D.C., 1992. The remarkable ability of turbulence model equations to describe particles in distinct element simulations. Chem. Eng. Sci. 60 (5), 1303–1312. ISSN transition. In: Proc. 5th Symp. on Numerical and Physical Aspects of Aerodynamic 0009-2509. Flows, January 13–15. California State University, Long Beach, California. Revil-Baudard, T., Chauchat, J., 2013. A two-phase model for sheet flow regime based on Wilcox, D.C., 1993. Comparison of two-equation turbulence models for boundary layers dense granular flow rheology. J. Geophys. Res.: Oceans 118 (2), 619–634. ISSN 2169- with pressure gradient. AIAA J. 31 (8), 1414–1421. ISSN 0001-1452. 9291. Wilson, K.C., 1987. Analysis of bed-load motion at high shear stress. J. Hydraul. Eng. 113 Revil-Baudard, T., Chauchat, J., Hurther, D., Barraud, P.-A., 2015. Investigation of sheet- (1), 97–103. ISSN 0733-9429. flow processes based on novel acoustic high-resolution velocity and concentration Zhou, Z., Kuang, S., Chu, K., Yu, A., 2010. Discrete particle simulation of particle fluid measurements. J. Fluid Mech. 767, 1–30. ISSN 1469-7645. flow: model formulations and their applicability. J. Fluid Mech. 661, 482–510. ISSN Revil-Baudard, T., Chauchat, J., Hurther, D., Eiff, O., 2016. Turbulence modifications 1469-7645.

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